Theorem elz12s 28464

Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
elz12s	⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elz12s

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2		id 22	. . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦)))
3		ovex 7400	. . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V
4	2, 3	eqeltrdi 2844	. . . 4 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V))
6	5	rexlimivv 3179	. 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
7		eqeq1 2740	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦))))
8	7	2rexbidv 3202	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
9		df-z12s 28407	. . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))}
10	8, 9	elab2g 3623	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
11	1, 6, 10	pm5.21nii 378	1 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429  (class class class)co 7367   /_su cdivs 28179  ℕ_0scn0s 28304  ℤ_sczs 28370  2_sc2s 28402  ↑_scexps 28404  ℤ_s[1/2]cz12s 28406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rex 3062  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-sn 4568  df-pr 4570  df-uni 4851  df-iota 6454  df-fv 6506  df-ov 7370  df-z12s 28407

This theorem is referenced by:  elz12si  28465  zz12s  28467  z12no  28468  z12addscl  28469  z12negscl  28470  z12shalf  28472  z12zsodd  28474  z12sge0  28475